Vol 50, No 2 (2016)

A uniqueness theorem for a convolution equation is proved for a class of infinitedimensional spaces larger than the class of Banach spaces, in particular, for L_p-spaces with p > 0.

The boundary of the Gelfand–Tsetlin graph is an infinite-dimensional locally compact space whose points parameterize the extreme characters of the infinite-dimensional group U(∞). The problem of harmonic analysis on the group U(∞) leads to a continuous family of probability measures on the boundary—the so-called zw-measures. Recently Vadim Gorin and the author have begun to study a q-analogue of the zw-measures. It turned out that constructing them requires introducing a novel combinatorial object, the extended Gelfand–Tsetlin graph. In the present paper it is proved that the Markov kernels connected with the extended Gelfand–Tsetlin graph and its q-boundary possess the Feller property. This property is needed for constructing a Markov dynamics on the q-boundary. A connection with the B-splines and their q-analogues is also discussed.

Let M be a von Neumann algebra equipped with a normal finite faithful normalized trace τ, and let A be a tracial subalgebra of M. Let E be a symmetric quasi-Banach space on [0, 1]. We show that A has an L_E(M)-factorization if and only if A is a subdiagonal algebra.

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ℂ is a variable and q ∈ ℂ, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.

It is shown that the group PSL₂(ℝ) is a spherical subgroup in the group of C³-diffeomorphisms of the circle.

To an arbitrary involutive stereotype algebra A the continuous envelope operation assigns its nearest, in some sense, involutive stereotype algebra En_vCA so that homomorphisms to various C*-algebras separate the elements of Env_C A but do not distinguish between the properties of A and those of Env_CA.

If A is an involutive stereotype subalgebra in the algebra C(M) of continuous functions on a paracompact locally compact topological space M, then, for C(M) to be a continuous envelope of A, i.e., Env_CA = C(M), it is necessary but not sufficient that A be dense in C(M). In this note we announce a necessary and sufficient condition for this: the involutive spectrum of A must coincide with M up to a weakening of the topology such that the system of compact subsets in M and the topology on each compact subset remains the same.